Why was the VC dimension not defined for all configurations of $d$ points?
I would say, that meaning of VC dimension is at least a possibility to implement any function on the given number of points for some case, not an express any function on $n$ points.
Yes, you are right, that this definition, unfortunately, is not very useful in practice.
Say, family of functions $\text{sign}(\sin(ax))$ has infinite $\text{VC}$ dimension. There exists an infinite sequence $x_n$, such that by tuning the parameter $a$ one can implement any logical function $0 1 0 \ldots 1$ on these points. However, this doesn't make this class of functions an outstanding ML algorithm.